The dimension of posets with planar cover graphs excluding two long incomparable chains

It has been known for more than 40 years that there are posets with planar cover graphs and arbitrarily large dimension. Recently, Streib and Trotter proved that such posets must have large height. In fact, all known constructions of such posets have two large disjoint chains with all points in one chain incomparable with all points in the other. Gutowski and Krawczyk conjectured that this feature is necessary. More formally, they conjectured that for every k > 1, there is a constant d such that if P is a poset with planar cover graph and P excludes k + k, then dim(P ) 6 d. We settle their conjecture in the affirmative. The proof involves some intermediate results that we believe are of independent interest.